Nonlinear Dynamics : міжнародна конференція

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Bushes of Nonlinear Normal Modes in Single-Layer Graphene
(NTU "KhPI", 2016) Chechin, G. M.; Ryabov, D. S.; Shcherbinin, S. A.
In-plane vibrations in uniformly stretched single-layer graphene (space group P6mm), which are described by the Rosenberg nonlinear normal modes (NNMs) and their bushes, are studied with the aid of group-theoretical methods developed by authors in some earlier papers. It was found that only 4 symmetry-determined NNMs (one-dimensional bushes), as well as 14 two-dimensional bushes are possible in graphene. They are exact solutions to the dynamical equations of this two-dimensional crystal. The verification of group-theoretical results with the aid of ab initio simulations based on density functional theory is discussed.
Construction of Nonlinear Normal Modes by Shaw-Pierre via Schur Decomposition
(NTU "KhPI", 2016) Perepelkin, Nikolay V.
In the paper the simplification of construction of nonlinear normal vibration modes by Shaw-Pierre in power series form is considered. The simplification can be obtained via change of variables in the equations o f motion of dynamical system under consideration. This change of variables is constructed by means of so-called ordered Schur matrix decomposition. As the result of the transformation there is no need in solving nonlinear algebraic equations in order to evaluate coefficients of nonlinear normal mode.
Resonance Behavior of the Forced Dissipative Spring-Pendulum System
(NTU "KhPI", 2016) Plaksiy, Kateryna Yu.; Mikhlin, Yuri V.
Dynamics of the dissipative spring-pendulum system under periodic external excitation in the vicinity of external resonance and simultaneous external and internal resonances is studied. Analysis of the system resonance behaviour is made on the base of the concept of nonlinear normal vibration modes (NNMs), which is generalized for systems with small dissipation. The multiple scales method and subsequent transformation to the reduced system with respect to the system energy, an arctangent of the amplitudes ratio and a difference of phases of required solutions are applied. Equilibrium positions of the reduced system correspond to nonlinear normal modes. So-called Transient nonlinear normal modes (TNNMs), which exist only for some certain levels of the system energy are selected. In the vicinity of values of time, corresponding to these energy levels, these TNNMs temporarily attract other system motions. Interaction of nonlinear vibration modes under resonance conditions is also analysed. Reliability of obtained analytical results is confirmed by numerical and numerical-analytical simulation.
Semi-Inverse Method in the Nonlinear Dynamics
(NTU "KhPI", 2016) Manevitch, Leonid I.; Smirnov, Valeri V.
The semi-inverse method based on using an internal small parameter of the nonlinear problems is discussed on the examples of the chain of coupled pendula and of the forced pendulum. The efficiency of such approach is highly appeared when the non-stationary dynamical problems are considered. In the framework of this method we demonstrate that both the spectrum of nonlinear normal modes and the interaction of them can be analysed successfully.
Nonlinear Dynamic Analysis of Elastic Rotor with Disk on Cantilevered End Supported on Angular Contact Ball Bearings
(NTU "KhPI", 2016) Filipkovskiy, Sergey V.
The mathematical model of rotor nonlinear oscillations on angular contact ball-bearings has been developed. The disk is fixed on the console end of the shaft. The shaft deflection and the elastic deformation of the bearings have the same order. The free oscillations have been analyzed by nonlinear normal modes. The modes and backbone curves of rotor nonlinear oscillations have been calculated. Oscillations are excited by the simultaneous action of the rotor unbalance and vibration of supports. The frequency response, orbits and Poincare maps have been constructed on the mode when rotating speed of the rotor is in the frequency range of supports vibration. The analysis o f nonlinear dynamics of the rotor has shown that besides the main resonance at low frequencies there are superresonant oscillations. The unstable modes saddle-node bifurcations leading to beats are observed.